Irreducibility and joint contrallability observability in singular systems

The concept of irreducibility of MFD´s at infinity is introduced. By merely considering common divisors at infinity, this concept turns out to be a complement of the well-known irreducibility for MFD´s. A combination of these two naturally leads to the notion of complete irreducibility. The second part of the paper is devoted to show that for a so-called pseudo-proper MFD, a block controllable singular system realization can be obtained such that (1). the order of the realized singular system is equal to the generalized determinantal degree of the denominator (GDDD) of this MFD; (2). the realization is controllable; (3). it will be observable if and only if the MFD is completely irreducible. In the final part of the paper, we show that for an arbitrary MFD, any of its nth-order singular system realization (n is GDDD) will be jointly controllable and observable if and only if the MFD is completely irreducible, which is an interesting counterpart of the case for regular systems.